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The three lines connecting the excenters of the given triangle to the corresponding edge midpoints all meet at the mittenpunkt; thus, it is the
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418:{\displaystyle \cot {\frac {A}{2}}:\cot {\frac {B}{2}}:\cot {\frac {C}{2}}=(\csc A+\cot A):(\csc B+\cot B):(\csc C+\cot C).}
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Kimberling, Clark (1994), "Central Points and
Central Lines in the Plane of a Triangle",
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Paul Yiu, "The uses of homogeneous barycentric coordinates in plane euclidean geometry"
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are the side lengths of the given triangle. Expressed instead in terms of the angles
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http://lya.fciencias.unam.mx/gfgf/ga20071/data/material/barycentricpaper.pdf
598:{\displaystyle a(b+c-a):b(c+a-b):c(a+b-c)=(1+\cos A):(1+\cos B):(1+\cos C).}
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Triangle center: symmedian point of the triangle's excentral triangle
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Untersuchungen über die wichtigsten zum
Dreiecke gehörenden Kreise
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of the given triangle, the ellipse tangent to the triangle at its
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The mittenpunkt is at the intersection of the line connecting the
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of the excentral triangle and the median triangle, with the
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http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
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800:"A Desarguesian dual for Nagel's middlespoint"
119:of the triangle. It was identified in 1836 by
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224:{\displaystyle (b+c-a):(c+a-b):(a+b-c)}
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54:to each corresponding edge midpoint (
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681:The Mittenpunkt also serves as the
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657:being the trilinear polar of the
655:corresponding axis of perspective
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776:Encyclopedia of Triangle Centers
50: Lines from the triangle's
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661:. The mittenpunkt is also the
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831:"Generalized Mandart conics"
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641:involving the mittenpunkt.
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637:, thus establishing three
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117:Euclidean transformations
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829:Gibert, Bernard (2004),
798:Eddy, Roland H. (1989),
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27: Arbitrary triangle
804:Elemente der Mathematik
430:barycentric coordinates
131:of the given triangle.
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651:center of perspective
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258:, the trilinears are
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141:trilinear coordinates
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711:Mathematics Magazine
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139:The mittenpunkt has
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869:Weisstein, Eric W.
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129:excentral triangle
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667:Mandart inellipse
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37:(centered at the
35:Mandart inellipse
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627:symmedian point
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717:(3): 163–187,
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683:Gergonne point
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635:Spieker center
619:Gergonne point
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105:middle point
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844:: 177–198,
631:orthocenter
135:Coordinates
95:mittenpunkt
77:Nagel point
60:mittenpunkt
39:mittenpunkt
693:References
56:concurrent
877:MathWorld
759:, Leipzig
633:with the
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512:−
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213:−
189:−
165:−
125:symmedian
73:Splitters
52:excenters
890:Category
755:(1836),
663:centroid
625:and the
623:incenter
617:and the
615:centroid
109:triangle
89:geometry
850:2130231
816:0999636
739:1573021
731:2690608
685:of the
665:of the
123:as the
107:) of a
58:at the
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254:, and
242:, and
234:where
101:German
99:(from
91:, the
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33:
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25:
834:(PDF)
727:JSTOR
677:Notes
111:is a
432:are
428:The
719:doi
581:cos
557:cos
533:cos
401:cot
389:csc
371:cot
359:csc
341:cot
329:csc
307:cot
288:cot
269:cot
87:In
892::
874:.
846:MR
840:,
836:,
812:MR
808:44
806:,
802:,
766:^
735:MR
733:,
725:,
715:67
713:,
701:^
689:.
673:.
250:,
238:,
103::
880:.
853:.
842:4
819:.
761:.
742:.
721::
593:.
590:)
587:C
578:+
575:1
572:(
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563:B
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548:(
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527:1
524:(
521:=
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515:c
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500:(
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473:(
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467::
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446:(
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413:.
410:)
407:C
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386:(
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377:B
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365:B
356:(
353::
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347:A
338:+
335:A
326:(
323:=
318:2
315:C
304::
299:2
296:B
285::
280:2
277:A
256:C
252:B
248:A
244:c
240:b
236:a
219:)
216:c
210:b
207:+
204:a
201:(
198::
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192:b
186:a
183:+
180:c
177:(
174::
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168:a
162:c
159:+
156:b
153:(
82:)
80:N
65:)
63:M
44:)
42:M
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